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13    EXTENSIONS OF HERMITIAN OPERATORS       443

it is necessary and sufficient that/satisfy the following two conditions : (1) the sequence
(/(0Vi)/o-n) is bounded; (2) the function

on S x S is of positive type. (To prove sufficiency, note that there exists a separable
Hilbert space E and a mapping si— >us of S into E such that k(s, t) = (us \ ut) (Section
6.3, Problem 8(c)), and we may even assume that the set of points us is total in E.
Show then that the sequence (uan) in E is bounded and converges weakly to a point
UQ (use (12.15.7.1) and (7.5.5)); we have (ut\u0) =f(s)/s for all s e S. Show that there
exists an unbounded hermitian operator H on E such that H- us = (us — UQ)/S for all
s e S; using Problem 14, show that there exists a Hilbert space F containing E, and
an unbounded self-adjoint operator A on F such that, for J1 ' z > 0, the function
F(z) = z((I — zA)~l - UQ | UQ) satisfies the required conditions* Putting

show that J^F(z) = */(z||i>2||2) and that ||uz|| sin 9? g ||w0|| for z e C(<p).) y).